Nash Equilibria of Noncooperative/Mixed Differential Games with Density Constraints in Infinite Dimensions

TL;DR

This paper develops a theoretical framework for analyzing Nash equilibria in infinite-dimensional differential games with density constraints, inspired by Cournot models, and validates it through numerical simulations.

Contribution

It introduces novel models for noncooperative and cooperative differential games with density constraints in infinite dimensions, establishing conditions for equilibrium uniqueness and computational reduction.

Findings

Established relationship between differential games and variational inequalities.

Proved conditions for uniqueness of equilibria.

Validated models with numerical simulations.

Abstract

Motivated by Cournot models, this paper proposes novel models of the noncooperative and cooperative differential games with density constraints in infinite dimensions, where markets consist of infinite firms and demand dynamics are governed by controlled differential equations. Markets engage in noncooperative competition with each other, while firms within each market engage in noncooperative or cooperative games. The main problems are to find the noncooperative Nash equilibrium (NNE) of the noncooperative differential game and the mixed Nash equilibrium (MNE) of the mixed noncooperative and cooperative differential game. Moreover, fundamental relationship is established between noncooperative/mixed differential game with density constraints and infinite-dimensional differential variational inequalities with density constraints. By variational analysis, it is proved under two conditions with certain symmetry that both of the two equilibrium problems can be reduced to solving systems of finite-dimensional projection equations with integral constraints by iterative computational methods. Crucially, the two conditions with certain symmetry, ensuring the uniqueness of the NNE and the MNE, provide theoretical foundations for strategic decision making regarding competitive versus cooperative market behaviors. Finally, the theoretical framework is validated through numerical simulations demonstrating the efficacy of our results.